ARIMAX/SARIMAX/VAR

Multi-Variable Dataset, Model Fitting, Diagnostics, Forcasting, and Validation

Literature Review

We will conduct a brief analysis on the topics of SARIMAX, ARIMAX, and VAR models in the context of monetary policy and treasury markets. Specifically, we will explore the relationships between inflation, the Federal Funds Rate, 30-Year Treasury Yield, and other related variables.

SARIMAX Model for Inflation: The Seasonal Autoregressive Integrated Moving Average with Exogenous Variables model is a powerful tool for time series analysis that incorporates external or exogenous variables into the traditional ARIMA framework. In the context of inflation forecasting, the SARIMAX model allows us to consider how variables like the Federal Funds Rate and lagged Federal Funds Rate affect inflation dynamics.

  • Federal Funds Rate: Research has shown that changes in the Federal Funds Rate, which is controlled by the Federal Reserve, can have a significant impact on inflation. A higher Federal Funds Rate tends to put downward pressure on inflation by making borrowing more expensive. Conversely, a lower rate can stimulate economic activity and potentially lead to higher inflation.

ARIMAX Model for 30-Year Treasury Yield: The Autoregressive Integrated Moving Average with Exogenous Variables model is an extension of the ARIMA model that incorporates external variables. When applied to the 30-Year Treasury Yield, it allows us to assess the influence of the Federal Funds Rate on long-term interest rates.

  • Federal Funds Rate: Changes in the Federal Funds Rate can influence the 30-Year Treasury Yield. An increase in the Federal Funds Rate typically leads to higher long-term interest rates, including the 30-Year Treasury Yield. This relationship is essential for understanding the impact of monetary policy on the bond market.

VAR Model for Monetary Policy and Treasury Markets: Vector Autoregression (VAR) models are used to analyze the dynamic relationships among multiple time series variables. In the context of monetary policy and treasury markets, a VAR model can help us understand how different variables, including the Federal Funds Rate and various Treasury Yields, interact with each other.

  • Fed Funds Rate, Treasury Yields: A VAR model that includes variables such as the Federal Funds Rate, 3-Month Yield, 6-Month Yield, 1-Year Yield, 5-Year Yield, 10-Year Yield, and 30-Year Yield can provide insights into the interconnectedness of these financial indicators. For example, an increase in the Federal Funds Rate can have ripple effects on short-term and long-term yields, impacting borrowing costs and investment decisions.

In summary, the literature review suggests that the Federal Funds Rate is a central variable that affects both inflation and interest rates. SARIMAX and ARIMAX models can be employed to capture these relationships, while a VAR model can provide a comprehensive view of the interactions among various monetary policy and treasury market variables. These modeling approaches are valuable tools for forecasting and policy analysis in the realm of monetary policy and treasury markets.

SARIMAX - Inflation vs. Fed Funds Rate

Data Processing

The plot shows the CPI and the Federal Funds Rate over time. The CPI has experienced a steady increase suggesting inflation growth, while the Federal Funds Rate has remained relatively stable, with only minor fluctuations over the same period.


Stationary Check

Augmented Dickey-Fuller Test Results for FedFundsRate :
Test Statistic: -8.408184 
P-value: 0.01 
Critical Values:  
The time series FedFundsRate is stationary based on the ADF test.

Augmented Dickey-Fuller Test Results for L2FedFundsRate :
Test Statistic: -8.367854 
P-value: 0.01 
Critical Values:  
The time series L2FedFundsRate is stationary based on the ADF test.

Augmented Dickey-Fuller Test Results for CPI :
Test Statistic: -5.128259 
P-value: 0.01 
Critical Values:  
The time series CPI is stationary based on the ADF test.


Model Fitting

auto.arima

auto.arima suggests a ARIMA model with seasonal components: SARIMAX(0,1,3)(0,0,2)[12]

Series: ts_combined[, "CPI"] 
Regression with ARIMA(0,1,3)(0,0,2)[12] errors 

Coefficients:
          ma1      ma2      ma3     sma1     sma2      l0      l2
      -0.4390  -0.3706  -0.0703  -0.1380  -0.0611  0.0279  0.0036
s.e.   0.0359   0.0350   0.0360   0.0361   0.0365  0.0291  0.0291

sigma^2 = 0.1436:  log likelihood = -367.37
AIC=750.74   AICc=750.92   BIC=788.47

Training set error measures:
                     ME      RMSE       MAE MPE MAPE      MASE         ACF1
Training set 0.01415216 0.3770653 0.2166059 NaN  Inf 0.6033404 -0.004461179


Custom Model

The first regression model shows both the current and lagged Federal Funds Rate positively affecting the Consumer Price Index (CPI), with a slightly better fit than the second model. The second model, using only the current Federal Funds Rate, also indicates a positive effect on CPI but with a weaker fit. Both models confirm a significant relationship between the Federal Funds Rate and CPI.

1. CPI ~ FedFundsRate + L2FedFundsRate


Call:
lm(formula = CPI ~ FedFundsRate + L2FedFundsRate, data = ts_combined)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.1340 -0.2589 -0.0540  0.1853  3.0811 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     0.33772    0.01681  20.085   <2e-16 ***
FedFundsRate    0.07781    0.03446   2.258   0.0242 *  
L2FedFundsRate  0.05750    0.03444   1.670   0.0953 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4832 on 823 degrees of freedom
Multiple R-squared:  0.008938,  Adjusted R-squared:  0.00653 
F-statistic: 3.711 on 2 and 823 DF,  p-value: 0.02486

2. CPI ~ FedFundsRate


Call:
lm(formula = CPI ~ FedFundsRate, data = ts_combined)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.1371 -0.2586 -0.0499  0.1714  3.0894 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   0.33804    0.01683   20.08   <2e-16 ***
FedFundsRate  0.07401    0.03442    2.15   0.0318 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4837 on 824 degrees of freedom
Multiple R-squared:  0.005581,  Adjusted R-squared:  0.004374 
F-statistic: 4.624 on 1 and 824 DF,  p-value: 0.03181


ACF & PACF Plot

Significant lags shown in the first difference ACF and PACF plots: d=0,1, D=0,1, P=1,2, Q=1, p=1,2, q=0,1


Hyperparameter Optimization

Model fitting with minimum AIC:
 1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984

Model fitting with minimum AICc:
 1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984

Model fitting with minimum BIC:
 1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984


Model Diagnostics

Both SARIMAX(1,0,2)(2,1,1)[12] and SARIMAX(0,1,3)(0,0,2)[12] evaluated through their diagnostic plots and statistical outputs suggest adequate fits for time series forecasting. The first model’s autocorrelations are within acceptable limits, and key coefficients are statistically significant, indicating a good fit, albeit with a non-significant constant term. The second model presents a marginally better fit, as evidenced by lower AIC and BIC values, and most coefficients are significant, though ma2 and sma2 are not. Both models have residuals that are approximately normally distributed and independent according to the Ljung-Box test, with the second model potentially being the preferred choice due to its lower complexity and better fit indicators.

 [1] "Call:"                                                                                         
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "        
 [3] "    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
 [4] "        REPORT = 1, reltol = tol))"                                                            
 [5] ""                                                                                              
 [6] "Coefficients:"                                                                                 
 [7] "         ar1      ma1      ma2     sar1     sar2     sma1  constant"                           
 [8] "      0.1529  -0.5980  -0.3007  -0.1782  -0.1219  -0.9520         0"                           
 [9] "s.e.  0.0808   0.0773   0.0642   0.0396   0.0406   0.0246         0"                           
[10] ""                                                                                              
[11] "sigma^2 estimated as 0.1438:  log likelihood = -383.89,  aic = 783.78"                         
[12] ""                                                                                              
[13] "$degrees_of_freedom"                                                                           
[14] "[1] 806"                                                                                       
[15] ""                                                                                              
[16] "$ttable"                                                                                       
[17] "         Estimate     SE  t.value p.value"                                                     
[18] "ar1        0.1529 0.0808   1.8927  0.0588"                                                     
[19] "ma1       -0.5980 0.0773  -7.7396  0.0000"                                                     
[20] "ma2       -0.3007 0.0642  -4.6855  0.0000"                                                     
[21] "sar1      -0.1782 0.0396  -4.4990  0.0000"                                                     
[22] "sar2      -0.1219 0.0406  -3.0068  0.0027"                                                     
[23] "sma1      -0.9520 0.0246 -38.7447  0.0000"                                                     
[24] "constant   0.0000 0.0000   0.1604  0.8726"                                                     
[25] ""                                                                                              
[26] "$AIC"                                                                                          
[27] "[1] 0.9640587"                                                                                 
[28] ""                                                                                              
[29] "$AICc"                                                                                         
[30] "[1] 0.9642298"                                                                                 
[31] ""                                                                                              
[32] "$BIC"                                                                                          
[33] "[1] 1.010314"                                                                                  
[34] ""                                                                                              

 [1] "Call:"                                                                                         
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "        
 [3] "    xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
 [4] "        REPORT = 1, reltol = tol))"                                                            
 [5] ""                                                                                              
 [6] "Coefficients:"                                                                                 
 [7] "          ma1     ma2     ma3     sma1     sma2  constant"                                     
 [8] "      -1.4737  0.0731  0.4006  -0.1446  -0.0540         0"                                     
 [9] "s.e.   0.0318  0.0570  0.0313   0.0359   0.0364         0"                                     
[10] ""                                                                                              
[11] "sigma^2 estimated as 0.1433:  log likelihood = -375.27,  aic = 764.54"                         
[12] ""                                                                                              
[13] "$degrees_of_freedom"                                                                           
[14] "[1] 818"                                                                                       
[15] ""                                                                                              
[16] "$ttable"                                                                                       
[17] "         Estimate     SE  t.value p.value"                                                     
[18] "ma1       -1.4737 0.0318 -46.2772  0.0000"                                                     
[19] "ma2        0.0731 0.0570   1.2818  0.2003"                                                     
[20] "ma3        0.4006 0.0313  12.7825  0.0000"                                                     
[21] "sma1      -0.1446 0.0359  -4.0336  0.0001"                                                     
[22] "sma2      -0.0540 0.0364  -1.4845  0.1381"                                                     
[23] "constant   0.0000 0.0000   0.1387  0.8897"                                                     
[24] ""                                                                                              
[25] "$AIC"                                                                                          
[26] "[1] 0.927841"                                                                                  
[27] ""                                                                                              
[28] "$AICc"                                                                                         
[29] "[1] 0.9279657"                                                                                 
[30] ""                                                                                              
[31] "$BIC"                                                                                          
[32] "[1] 0.9678885"                                                                                 
[33] ""                                                                                              


Cross Validation

The SARIMAX(0,1,3)(0,0,2)[12] model demonstrates superior performance in cross-validation, exhibiting lower RMSE values during testing phases. This indicates a more accurate and reliable forecasting ability, particularly in capturing the seasonal trends and dependencies in the data.


Forecasting

The SARIMA model forecasts for the Federal Funds Rate and CPI suggest upcoming economic uncertainty. The Fed Funds Rate prediction shows potential for slight increases and a broad range of outcomes, signaling that interest rate policies may need to adapt to evolving economic conditions. The CPI forecast indicates rising volatility, with a trend that suggests inflation could be a concern, requiring careful monitoring and possible intervention from policymakers.

ARIMAX - 30-Year Treasury Yield vs. Fed Funds Rate

Data Processing

The plot shows the 30-Year Treasury Yield and the Federal Funds Rate over time. Both series appear to move together over the years, suggesting a relationship where changes in the Federal Funds Rate may be associated with similar movements in the long-term Treasury Yield.


Stationary Check

Augmented Dickey-Fuller Test Results for FedFundsRate :
Test Statistic: -6.327969 
P-value: 0.01 
Critical Values:  
The time series FedFundsRate is stationary based on the ADF test.

Augmented Dickey-Fuller Test Results for treasury_yield_30 :
Test Statistic: -7.3842 
P-value: 0.01 
Critical Values:  
The time series treasury_yield_30 is stationary based on the ADF test.


Model Fitting

auto.arima

auto.arima suggests a ARIMA model: ARIMAX(0,0,2)

Series: ts_combined[, 2] 
Regression with ARIMA(0,0,2) errors 

Coefficients:
         ma1      ma2    xreg
      0.3316  -0.1327  0.1263
s.e.  0.0437   0.0429  0.0217

sigma^2 = 0.05685:  log likelihood = 9.65
AIC=-11.29   AICc=-11.22   BIC=6.01

Training set error measures:
                       ME      RMSE       MAE       MPE     MAPE      MASE
Training set -0.005267137 0.2377827 0.1712959 -2.100979 260.2457 0.6323817
                    ACF1
Training set 0.001726696


Custom Model


Call:
lm(formula = treasury_yield_30 ~ FedFundsRate, data = ts_combined)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.09369 -0.13309 -0.00534  0.12427  1.49825 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -0.006401   0.010703  -0.598     0.55    
FedFundsRate  0.149370   0.020198   7.395 5.23e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2528 on 556 degrees of freedom
Multiple R-squared:  0.08955,   Adjusted R-squared:  0.08791 
F-statistic: 54.69 on 1 and 556 DF,  p-value: 5.23e-13


ACF & PACF Plot

Significant lags shown in the first difference ACF and PACF plots: d=0,1, p=1,2, q=1,2


Hyperparameter Optimization

Model fitting with minimum AIC:
 1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554

Model fitting with minimum AICc:
 1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554

Model fitting with minimum BIC:
 1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554


Model Diagnostics

The ARIMA model diagnostics for the 30-year Treasury yields and the Federal Funds Rate indicate effective fits with well-behaved residuals, suggesting that the models capture the essential patterns of these financial indicators. The significant terms in each model reflect the distinct behaviors of the yields and rates, with the 30-year Treasury likely exhibiting more long-term trends and the Federal Funds Rate responding to short-term influences. The slightly superior AIC and AICc values for the second model hint at a better fit for capturing the quick adjustments typical in the Federal Funds Rate.

 [1] "Call:"                                                                                 
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
 [3] "    xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed, "       
 [4] "    optim.control = list(trace = trc, REPORT = 1, reltol = tol))"                      
 [5] ""                                                                                      
 [6] "Coefficients:"                                                                         
 [7] "          ar1     ma1   xmean"                                                         
 [8] "      -0.2930  0.6257  0.0001"                                                         
 [9] "s.e.   0.0868  0.0686  0.0127"                                                         
[10] ""                                                                                      
[11] "sigma^2 estimated as 0.0568:  log likelihood = 8.38,  aic = -8.75"                     
[12] ""                                                                                      
[13] "$degrees_of_freedom"                                                                   
[14] "[1] 555"                                                                               
[15] ""                                                                                      
[16] "$ttable"                                                                               
[17] "      Estimate     SE t.value p.value"                                                 
[18] "ar1    -0.2930 0.0868 -3.3746  0.0008"                                                 
[19] "ma1     0.6257 0.0686  9.1212  0.0000"                                                 
[20] "xmean   0.0001 0.0127  0.0112  0.9911"                                                 
[21] ""                                                                                      
[22] "$AIC"                                                                                  
[23] "[1] -0.01568965"                                                                       
[24] ""                                                                                      
[25] "$AICc"                                                                                 
[26] "[1] -0.01561201"                                                                       
[27] ""                                                                                      
[28] "$BIC"                                                                                  
[29] "[1] 0.01530934"                                                                        
[30] ""                                                                                      

 [1] "Call:"                                                                                 
 [2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
 [3] "    xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed, "       
 [4] "    optim.control = list(trace = trc, REPORT = 1, reltol = tol))"                      
 [5] ""                                                                                      
 [6] "Coefficients:"                                                                         
 [7] "         ma1      ma2   xmean"                                                         
 [8] "      0.3192  -0.1435  0.0001"                                                         
 [9] "s.e.  0.0422   0.0418  0.0118"                                                         
[10] ""                                                                                      
[11] "sigma^2 estimated as 0.05663:  log likelihood = 9.22,  aic = -10.45"                   
[12] ""                                                                                      
[13] "$degrees_of_freedom"                                                                   
[14] "[1] 555"                                                                               
[15] ""                                                                                      
[16] "$ttable"                                                                               
[17] "      Estimate     SE t.value p.value"                                                 
[18] "ma1     0.3192 0.0422  7.5680  0.0000"                                                 
[19] "ma2    -0.1435 0.0418 -3.4368  0.0006"                                                 
[20] "xmean   0.0001 0.0118  0.0073  0.9941"                                                 
[21] ""                                                                                      
[22] "$AIC"                                                                                  
[23] "[1] -0.01871965"                                                                       
[24] ""                                                                                      
[25] "$AICc"                                                                                 
[26] "[1] -0.01864201"                                                                       
[27] ""                                                                                      
[28] "$BIC"                                                                                  
[29] "[1] 0.01227934"                                                                        
[30] ""                                                                                      


Cross Validation

The ARIMAX(1,0,1) model demonstrates superior performance in cross-validation, exhibiting lower RMSE values during testing phases. This indicates a more accurate and reliable forecasting ability, particularly in capturing the seasonal trends and dependencies in the data.


Foreasting

The SARIMA model forecasts for the Federal Funds Rate and the 30-Year Treasury Yield depict stability in the near term, with an increasing range of outcomes as we move further into the future. The Fed Funds Rate is shown to have a stable past with minor fluctuations, but the forecast suggests uncertainty, implying that rates could either rise or fall. Similarly, the Treasury Yield forecast indicates relatively stable past rates but projects increasing uncertainty. This suggests that the long-term lending market could face varied conditions, potentially affecting investment, inflation, and economic growth expectations.

VAR - Monetary Policy and Treasury Markets

Variables: Fed Funds Rate, 3-Month Yield, 6-Month Yield, 1-Year Yield, 5-Year Yield, 10-Year Yield, 30-Year Yield.

Data Processing

This graph illustrates the Federal Funds Rate (FFR) and various Treasury yields (ranging from 3 months to 30 years) over time. The general trend shows that Treasury yields across different maturities tend to move in the same direction as the Federal Funds Rate, with the highest yields typically associated with longer maturities.

The pairplot indicates a generally positive correlation among all interest rate variables, from the Fed Funds Rate to the 30-Year Yield—when one goes up, others often do too. The time series plot shows how these rates have moved over time, sometimes in sync, reflecting similar economic forces, and sometimes diverging due to different market expectations for the short and long term.


Model Fitting

VAR Selection

VAR Model Lag Value Selection:
AIC(n)  HQ(n)  SC(n) FPE(n) 
     2      2      2      2 


VAR(1)

The VAR(1) model analysis underscores the significant economic impact of the Federal Funds Rate (FFR) on various Treasury yields. It shows how changes in the FFR ripple through both short-term and long-term interest rates, influencing the entire yield curve. Short-term rates respond quickly to monetary policy shifts, while long-term rates reflect market expectations about future economic trends. This interconnection highlights the far-reaching effects of central bank policies on different aspects of the economy, from consumer borrowing to long-term investments.


VAR Estimation Results:
========================= 
Endogenous variables: FFR, T3Mon, T6Mon, T1Yr, T5Yr, T10Yr, T30Yr 
Deterministic variables: both 
Sample size: 499 
Log Likelihood: 3050.254 
Roots of the characteristic polynomial:
0.9735 0.9544 0.8926 0.8926 0.7743 0.7743 0.6561
Call:
vars::VAR(y = ts_combined, p = 1, type = c("both"))


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.7746280  0.0844082   9.177  < 2e-16 ***
T3Mon.l1  0.3511073  0.1123396   3.125  0.00188 ** 
T6Mon.l1 -0.4857568  0.2165015  -2.244  0.02530 *  
T1Yr.l1   0.7110530  0.1472259   4.830 1.83e-06 ***
T5Yr.l1  -0.4046803  0.1083445  -3.735  0.00021 ***
T10Yr.l1  0.3498053  0.1625501   2.152  0.03189 *  
T30Yr.l1 -0.1491100  0.0916170  -1.628  0.10427    
const     0.3653373  0.1379056   2.649  0.00833 ** 
trend    -0.0005838  0.0002444  -2.389  0.01728 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2337 on 490 degrees of freedom
Multiple R-Squared: 0.995,  Adjusted R-squared: 0.9949 
F-statistic: 1.215e+04 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T3Mon: 
====================================== 
T3Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.0685659  0.0639188   1.073   0.2839    
T3Mon.l1  0.9409624  0.0850701  11.061   <2e-16 ***
T6Mon.l1 -0.3813387  0.1639475  -2.326   0.0204 *  
T1Yr.l1   0.1451754  0.1114880   1.302   0.1935    
T5Yr.l1   0.1128966  0.0820447   1.376   0.1694    
T10Yr.l1 -0.0870301  0.1230924  -0.707   0.4799    
T30Yr.l1  0.0067650  0.0693777   0.098   0.9224    
const    -0.0440213  0.1044301  -0.422   0.6735    
trend     0.0002047  0.0001851   1.106   0.2692    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.177 on 490 degrees of freedom
Multiple R-Squared: 0.7574, Adjusted R-squared: 0.7534 
F-statistic: 191.2 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T6Mon: 
====================================== 
T6Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.0641697  0.0668795   0.959 0.337788    
T3Mon.l1  0.3186130  0.0890105   3.579 0.000379 ***
T6Mon.l1  0.1069623  0.1715415   0.624 0.533223    
T1Yr.l1   0.3236682  0.1166521   2.775 0.005737 ** 
T5Yr.l1   0.0984928  0.0858450   1.147 0.251804    
T10Yr.l1 -0.0589513  0.1287940  -0.458 0.647358    
T30Yr.l1 -0.0020282  0.0725913  -0.028 0.977721    
const    -0.1137577  0.1092673  -1.041 0.298345    
trend     0.0003620  0.0001936   1.869 0.062189 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.1852 on 490 degrees of freedom
Multiple R-Squared: 0.7376, Adjusted R-squared: 0.7333 
F-statistic: 172.2 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T1Yr: 
===================================== 
T1Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.0826055  0.0763778   1.082 0.279990    
T3Mon.l1  0.3137411  0.1016520   3.086 0.002140 ** 
T6Mon.l1 -0.6752194  0.1959042  -3.447 0.000616 ***
T1Yr.l1   1.0984511  0.1332193   8.245 1.52e-15 ***
T5Yr.l1   0.1476077  0.0980369   1.506 0.132806    
T10Yr.l1 -0.0847203  0.1470856  -0.576 0.564884    
T30Yr.l1  0.0067212  0.0829008   0.081 0.935415    
const    -0.0983891  0.1247857  -0.788 0.430806    
trend     0.0003020  0.0002211   1.366 0.172658    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2115 on 490 degrees of freedom
Multiple R-Squared: 0.7854, Adjusted R-squared: 0.7819 
F-statistic: 224.1 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T5Yr: 
===================================== 
T5Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.0690664  0.1036273   0.666  0.50541    
T3Mon.l1  0.3976029  0.1379186   2.883  0.00411 ** 
T6Mon.l1 -0.7755880  0.2657974  -2.918  0.00369 ** 
T1Yr.l1   0.0803278  0.1807482   0.444  0.65694    
T5Yr.l1   1.1045613  0.1330138   8.304 9.89e-16 ***
T10Yr.l1 -0.1243113  0.1995617  -0.623  0.53363    
T30Yr.l1  0.0707768  0.1124776   0.629  0.52948    
const    -0.1467030  0.1693058  -0.866  0.38664    
trend     0.0002964  0.0003000   0.988  0.32372    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.287 on 490 degrees of freedom
Multiple R-Squared: 0.9163, Adjusted R-squared: 0.9149 
F-statistic: 670.3 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T10Yr: 
====================================== 
T10Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1   -0.0293720  0.0894883  -0.328    0.743    
T3Mon.l1  0.7075690  0.1191008   5.941 5.38e-09 ***
T6Mon.l1 -1.1130431  0.2295317  -4.849 1.67e-06 ***
T1Yr.l1   0.6360693  0.1560868   4.075 5.36e-05 ***
T5Yr.l1  -0.1352939  0.1148652  -1.178    0.239    
T10Yr.l1  1.0141433  0.1723333   5.885 7.40e-09 ***
T30Yr.l1  0.0074114  0.0971310   0.076    0.939    
const     0.2207933  0.1462055   1.510    0.132    
trend    -0.0003204  0.0002591  -1.237    0.217    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2478 on 490 degrees of freedom
Multiple R-Squared: 0.9934, Adjusted R-squared: 0.9933 
F-statistic:  9219 on 8 and 490 DF,  p-value: < 2.2e-16 


Estimation results for equation T30Yr: 
====================================== 
T30Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1   -0.0927739  0.0796491  -1.165 0.244674    
T3Mon.l1  0.6233762  0.1060058   5.881 7.57e-09 ***
T6Mon.l1 -0.9272946  0.2042948  -4.539 7.12e-06 ***
T1Yr.l1   0.5013582  0.1389251   3.609 0.000339 ***
T5Yr.l1  -0.1750598  0.1022359  -1.712 0.087472 .  
T10Yr.l1  0.1989505  0.1533853   1.297 0.195220    
T30Yr.l1  0.8773440  0.0864515  10.148  < 2e-16 ***
const     0.2855076  0.1301303   2.194 0.028703 *  
trend    -0.0003541  0.0002306  -1.536 0.125266    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2206 on 490 degrees of freedom
Multiple R-Squared: 0.9938, Adjusted R-squared: 0.9937 
F-statistic:  9842 on 8 and 490 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
            FFR     T3Mon    T6Mon     T1Yr     T5Yr   T10Yr   T30Yr
FFR    0.054637 -0.006841 -0.01050 -0.01507 -0.03161 0.01695 0.01364
T3Mon -0.006841  0.031331  0.02878  0.02718  0.02575 0.01369 0.01077
T6Mon -0.010501  0.028779  0.03430  0.03661  0.03848 0.02227 0.01700
T1Yr  -0.015067  0.027176  0.03661  0.04474  0.05183 0.03040 0.02336
T5Yr  -0.031615  0.025752  0.03848  0.05183  0.08235 0.04693 0.03773
T10Yr  0.016950  0.013694  0.02227  0.03040  0.04693 0.06141 0.05193
T30Yr  0.013640  0.010769  0.01700  0.02336  0.03773 0.05193 0.04865

Correlation matrix of residuals:
          FFR   T3Mon   T6Mon    T1Yr    T5Yr  T10Yr  T30Yr
FFR    1.0000 -0.1653 -0.2426 -0.3048 -0.4713 0.2926 0.2646
T3Mon -0.1653  1.0000  0.8779  0.7259  0.5070 0.3122 0.2758
T6Mon -0.2426  0.8779  1.0000  0.9345  0.7239 0.4853 0.4162
T1Yr  -0.3048  0.7259  0.9345  1.0000  0.8540 0.5800 0.5008
T5Yr  -0.4713  0.5070  0.7239  0.8540  1.0000 0.6600 0.5962
T10Yr  0.2926  0.3122  0.4853  0.5800  0.6600 1.0000 0.9500
T30Yr  0.2646  0.2758  0.4162  0.5008  0.5962 0.9500 1.0000


VAR(2)

The VAR(2) model’s analysis shows how changes in the FFR, a key monetary policy tool, not only have immediate impacts but also exert influence over longer periods through their interactions with short-term and long-term yields. These interdependencies between different segments of the yield curve are crucial for understanding market expectations and economic conditions.


VAR Estimation Results:
========================= 
Endogenous variables: FFR, T3Mon, T6Mon, T1Yr, T5Yr, T10Yr, T30Yr 
Deterministic variables: both 
Sample size: 498 
Log Likelihood: 3235.419 
Roots of the characteristic polynomial:
0.9678 0.9101 0.8833 0.8592 0.8073 0.8073 0.6482 0.3795 0.3541 0.2308 0.1858 0.1302 0.1302 0.01743
Call:
vars::VAR(y = ts_combined, p = 2, type = "both")


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.9521757  0.1352793   7.039 6.71e-12 ***
T3Mon.l1  0.3785554  0.1352109   2.800 0.005320 ** 
T6Mon.l1 -0.6044160  0.2550213  -2.370 0.018178 *  
T1Yr.l1   0.6883991  0.2052075   3.355 0.000857 ***
T5Yr.l1  -0.5577872  0.1724705  -3.234 0.001304 ** 
T10Yr.l1  0.7228538  0.2362035   3.060 0.002334 ** 
T30Yr.l1 -0.2982019  0.1544073  -1.931 0.054036 .  
FFR.l2   -0.0556846  0.1341871  -0.415 0.678343    
T3Mon.l2 -0.1512887  0.1367911  -1.106 0.269285    
T6Mon.l2  0.2900315  0.2558634   1.134 0.257551    
T1Yr.l2  -0.2381943  0.2076678  -1.147 0.251952    
T5Yr.l2   0.3657711  0.1735552   2.108 0.035588 *  
T10Yr.l2 -0.5464220  0.2367433  -2.308 0.021417 *  
T30Yr.l2  0.1849556  0.1528606   1.210 0.226886    
const     0.4594991  0.1243409   3.695 0.000245 ***
trend    -0.0007465  0.0002202  -3.390 0.000755 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2055 on 482 degrees of freedom
Multiple R-Squared: 0.9961, Adjusted R-squared: 0.996 
F-statistic:  8211 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T3Mon: 
====================================== 
T3Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    2.233e-01  1.154e-01   1.936   0.0535 .  
T3Mon.l1  1.106e+00  1.153e-01   9.590   <2e-16 ***
T6Mon.l1 -5.292e-01  2.175e-01  -2.433   0.0153 *  
T1Yr.l1  -6.771e-03  1.750e-01  -0.039   0.9692    
T5Yr.l1   2.921e-01  1.471e-01   1.986   0.0476 *  
T10Yr.l1 -1.645e-01  2.014e-01  -0.817   0.4144    
T30Yr.l1  2.998e-02  1.317e-01   0.228   0.8200    
FFR.l2   -1.560e-01  1.144e-01  -1.363   0.1735    
T3Mon.l2 -2.750e-01  1.167e-01  -2.358   0.0188 *  
T6Mon.l2  3.412e-01  2.182e-01   1.564   0.1185    
T1Yr.l2   5.042e-02  1.771e-01   0.285   0.7760    
T5Yr.l2  -1.663e-01  1.480e-01  -1.124   0.2616    
T10Yr.l2  7.890e-02  2.019e-01   0.391   0.6961    
T30Yr.l2 -3.163e-02  1.304e-01  -0.243   0.8084    
const     1.947e-02  1.060e-01   0.184   0.8544    
trend     7.458e-05  1.878e-04   0.397   0.6914    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.1752 on 482 degrees of freedom
Multiple R-Squared: 0.7659, Adjusted R-squared: 0.7586 
F-statistic: 105.1 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T6Mon: 
====================================== 
T6Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)   
FFR.l1    0.2086962  0.1211763   1.722  0.08567 . 
T3Mon.l1  0.3987475  0.1211149   3.292  0.00107 **
T6Mon.l1 -0.0179431  0.2284349  -0.079  0.93742   
T1Yr.l1   0.2984660  0.1838143   1.624  0.10509   
T5Yr.l1   0.2974574  0.1544902   1.925  0.05477 . 
T10Yr.l1 -0.2257951  0.2115789  -1.067  0.28642   
T30Yr.l1  0.1240212  0.1383101   0.897  0.37033   
FFR.l2   -0.1523969  0.1201978  -1.268  0.20545   
T3Mon.l2 -0.2123357  0.1225304  -1.733  0.08375 . 
T6Mon.l2  0.3498190  0.2291892   1.526  0.12758   
T1Yr.l2  -0.0819299  0.1860181  -0.440  0.65982   
T5Yr.l2  -0.2028815  0.1554618  -1.305  0.19251   
T10Yr.l2  0.1678978  0.2120625   0.792  0.42890   
T30Yr.l2 -0.1233282  0.1369246  -0.901  0.36820   
const    -0.0849029  0.1113782  -0.762  0.44626   
trend     0.0002823  0.0001972   1.431  0.15304   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.1841 on 482 degrees of freedom
Multiple R-Squared: 0.745,  Adjusted R-squared: 0.7371 
F-statistic:  93.9 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T1Yr: 
===================================== 
T1Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.2113325  0.1351125   1.564   0.1184    
T3Mon.l1  0.2431981  0.1350441   1.801   0.0723 .  
T6Mon.l1 -0.6036530  0.2547069  -2.370   0.0182 *  
T1Yr.l1   1.0483188  0.2049545   5.115 4.54e-07 ***
T5Yr.l1   0.3915911  0.1722578   2.273   0.0234 *  
T10Yr.l1 -0.3111313  0.2359123  -1.319   0.1878    
T30Yr.l1  0.2146394  0.1542169   1.392   0.1646    
FFR.l2   -0.1460338  0.1340216  -1.090   0.2764    
T3Mon.l2 -0.0598441  0.1366224  -0.438   0.6616    
T6Mon.l2  0.1270787  0.2555479   0.497   0.6192    
T1Yr.l2  -0.0178044  0.2074118  -0.086   0.9316    
T5Yr.l2  -0.2756804  0.1733412  -1.590   0.1124    
T10Yr.l2  0.2243337  0.2364514   0.949   0.3432    
T30Yr.l2 -0.1870179  0.1526721  -1.225   0.2212    
const    -0.1127772  0.1241876  -0.908   0.3643    
trend     0.0002820  0.0002199   1.282   0.2003    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2052 on 482 degrees of freedom
Multiple R-Squared: 0.8012, Adjusted R-squared: 0.795 
F-statistic: 129.5 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T5Yr: 
===================================== 
T5Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.1372860  0.1775727   0.773   0.4398    
T3Mon.l1  0.2149833  0.1774828   1.211   0.2264    
T6Mon.l1 -0.7292255  0.3347505  -2.178   0.0299 *  
T1Yr.l1   0.0805036  0.2693630   0.299   0.7652    
T5Yr.l1   1.4981225  0.2263912   6.617 9.75e-11 ***
T10Yr.l1 -0.5528881  0.3100496  -1.783   0.0752 .  
T30Yr.l1  0.4731419  0.2026808   2.334   0.0200 *  
FFR.l2   -0.1151528  0.1761389  -0.654   0.5136    
T3Mon.l2  0.0840939  0.1795571   0.468   0.6398    
T6Mon.l2  0.1514553  0.3358558   0.451   0.6522    
T1Yr.l2  -0.0106697  0.2725925  -0.039   0.9688    
T5Yr.l2  -0.4903611  0.2278150  -2.152   0.0319 *  
T10Yr.l2  0.4558447  0.3107581   1.467   0.1431    
T30Yr.l2 -0.3773127  0.2006505  -1.880   0.0606 .  
const    -0.1574989  0.1632145  -0.965   0.3350    
trend     0.0002555  0.0002890   0.884   0.3771    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2697 on 482 degrees of freedom
Multiple R-Squared: 0.927,  Adjusted R-squared: 0.9248 
F-statistic: 408.2 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T10Yr: 
====================================== 
T10Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.1913693  0.1548277   1.236 0.217055    
T3Mon.l1  0.5707830  0.1547494   3.688 0.000251 ***
T6Mon.l1 -1.0672439  0.2918729  -3.657 0.000284 ***
T1Yr.l1   0.4950770  0.2348608   2.108 0.035550 *  
T5Yr.l1   0.1751568  0.1973932   0.887 0.375333    
T10Yr.l1  0.7822002  0.2703359   2.893 0.003983 ** 
T30Yr.l1  0.3752108  0.1767197   2.123 0.034246 *  
FFR.l2   -0.1847856  0.1535776  -1.203 0.229487    
T3Mon.l2 -0.0432282  0.1565579  -0.276 0.782577    
T6Mon.l2  0.2244892  0.2928366   0.767 0.443693    
T1Yr.l2  -0.0270054  0.2376766  -0.114 0.909584    
T5Yr.l2  -0.2569851  0.1986346  -1.294 0.196369    
T10Yr.l2  0.1525009  0.2709536   0.563 0.573812    
T30Yr.l2 -0.3338779  0.1749495  -1.908 0.056930 .  
const     0.3023725  0.1423086   2.125 0.034114 *  
trend    -0.0005014  0.0002520  -1.990 0.047197 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2352 on 482 degrees of freedom
Multiple R-Squared: 0.994,  Adjusted R-squared: 0.9939 
F-statistic:  5364 on 15 and 482 DF,  p-value: < 2.2e-16 


Estimation results for equation T30Yr: 
====================================== 
T30Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.0248940  0.1385365   0.180 0.857470    
T3Mon.l1  0.4661740  0.1384664   3.367 0.000822 ***
T6Mon.l1 -0.8266782  0.2611616  -3.165 0.001647 ** 
T1Yr.l1   0.3821162  0.2101484   1.818 0.069636 .  
T5Yr.l1  -0.0020661  0.1766232  -0.012 0.990672    
T10Yr.l1 -0.0368552  0.2418907  -0.152 0.878964    
T30Yr.l1  1.3033321  0.1581250   8.242 1.61e-15 ***
FFR.l2   -0.0848315  0.1374179  -0.617 0.537312    
T3Mon.l2  0.0420729  0.1400847   0.300 0.764048    
T6Mon.l2  0.0559831  0.2620239   0.214 0.830905    
T1Yr.l2   0.0093640  0.2126679   0.044 0.964898    
T5Yr.l2  -0.1230581  0.1777340  -0.692 0.489037    
T10Yr.l2  0.1789105  0.2424435   0.738 0.460906    
T30Yr.l2 -0.4114694  0.1565411  -2.629 0.008850 ** 
const     0.3683984  0.1273347   2.893 0.003987 ** 
trend    -0.0005069  0.0002255  -2.248 0.025031 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2104 on 482 degrees of freedom
Multiple R-Squared: 0.9944, Adjusted R-squared: 0.9942 
F-statistic:  5664 on 15 and 482 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
            FFR     T3Mon    T6Mon     T1Yr     T5Yr   T10Yr   T30Yr
FFR    0.042220 -0.009279 -0.01068 -0.01265 -0.02491 0.01365 0.01039
T3Mon -0.009279  0.030703  0.02875  0.02781  0.02727 0.01307 0.01034
T6Mon -0.010676  0.028749  0.03388  0.03573  0.03701 0.02067 0.01588
T1Yr  -0.012650  0.027810  0.03573  0.04212  0.04704 0.02775 0.02153
T5Yr  -0.024911  0.027270  0.03701  0.04704  0.07275 0.04316 0.03511
T10Yr  0.013648  0.013072  0.02067  0.02775  0.04316 0.05530 0.04687
T30Yr  0.010394  0.010343  0.01588  0.02153  0.03511 0.04687 0.04428

Correlation matrix of residuals:
          FFR   T3Mon   T6Mon    T1Yr    T5Yr  T10Yr  T30Yr
FFR    1.0000 -0.2577 -0.2823 -0.3000 -0.4495 0.2824 0.2404
T3Mon -0.2577  1.0000  0.8914  0.7734  0.5770 0.3172 0.2805
T6Mon -0.2823  0.8914  1.0000  0.9459  0.7455 0.4775 0.4100
T1Yr  -0.3000  0.7734  0.9459  1.0000  0.8499 0.5750 0.4987
T5Yr  -0.4495  0.5770  0.7455  0.8499  1.0000 0.6805 0.6187
T10Yr  0.2824  0.3172  0.4775  0.5750  0.6805 1.0000 0.9472
T30Yr  0.2404  0.2805  0.4100  0.4987  0.6187 0.9472 1.0000


Cross Validation

The plots compare the cross-validation RMSE for VAR(1) and VAR(2) models across different financial variables over time. The consistently lower RMSE values for VAR(1) indicate that it provides a better fit or more accurate forecasts for these series.


Forecasting

The forecast plots generated from the VAR(1) model for various interest rates, including the Fed Funds Rate and various Treasury yields, present a detailed picture of expected movements and uncertainties within the financial market. These plots suggest that each interest rate may follow a different path in the future, yet they are all interrelated. Typically, the Fed Funds Rate, which is set by the Federal Reserve, influences the other yields, as it reflects the cost of borrowing funds overnight. Treasury yields of different maturities react to this benchmark rate but also incorporate market expectations for inflation and economic growth over various horizons. The plots show that while forecasts are made for each rate individually, the market’s future expectations for each can diverge due to different influencing factors, including policy decisions, economic outlook, and investor sentiment.

VAR - Monetary Policy and Macroeconomic Indicators

Variables: FFR, GDP, Unemployment Rate, CPI, 30-Year Mortgage Rate

Data Processing

The pair plot illustrates the disparity in scales among different economic indicators, making it difficult to discern relationships directly due to the dominance of variables like GDP. The time series plot reveals distinct trends over time, such as the steady increase in GDP and the cyclical nature of the unemployment rate, alongside the long-term rise in CPI and the decline in the mortgage rate.


Model Fitting

The progression from VAR(1) through VAR(3), VAR(7), to VAR(10) models shows an evolving understanding of the interrelationships among the key economic indicators: Federal Funds Rate, Gross Domestic Product, Unemployment Rate, Consumer Price Index, and 30-Year Mortgage Rate.

In all models, the persistent influence of FFR on MORT is evident, highlighting the impact of monetary policy on long-term interest rates. The GDP equations across these models consistently show its own past values as significant predictors, indicating economic growth’s self-sustaining nature.

As the lag order increases, the models reveal more complex dynamics. For example, the VAR(10) model captures longer-term influences, showing how variables like UMP and CPI impact others over extended periods. However, with higher-order VAR models, the risk of overfitting increases, and some coefficients may become statistically insignificant, suggesting that not all lagged values meaningfully contribute to the model.

The overall fit of the model improves with higher lag orders, as seen in the increasing R-squared values, but this comes with the trade-off of increased complexity and potential redundancy in lagged terms. The choice of the model should balance the complexity with interpretability and the specific requirements of the analysis.


VAR Selection

VAR Model Lag Value Selection:
AIC(n)  HQ(n)  SC(n) FPE(n) 
    10      7      3     10 


VAR(1)


VAR Estimation Results:
========================= 
Endogenous variables: FFR, GDP, UMP, CPI, MORT 
Deterministic variables: both 
Sample size: 624 
Log Likelihood: -4567.829 
Roots of the characteristic polynomial:
    1     1 0.9706 0.9706 0.9289
Call:
vars::VAR(y = ts_combined, p = 1, type = c("both"))


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   9.779e-01  1.676e-02  58.357   <2e-16 ***
GDP.l1   5.989e-05  5.117e-05   1.170    0.242    
UMP.l1  -2.315e-02  1.615e-02  -1.433    0.152    
CPI.l1   4.340e-03  5.346e-03   0.812    0.417    
MORT.l1 -5.592e-03  2.425e-02  -0.231    0.818    
const    3.984e-02  3.771e-01   0.106    0.916    
trend   -3.820e-03  2.524e-03  -1.514    0.131    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.5307 on 617 degrees of freedom
Multiple R-Squared: 0.9823, Adjusted R-squared: 0.9821 
F-statistic:  5711 on 6 and 617 DF,  p-value: < 2.2e-16 


Estimation results for equation GDP: 
==================================== 
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend 

         Estimate Std. Error t value Pr(>|t|)    
FFR.l1   1.360650   1.805933   0.753    0.451    
GDP.l1   1.000391   0.005515 181.407  < 2e-16 ***
UMP.l1   9.904724   1.740747   5.690 1.96e-08 ***
CPI.l1  -0.793751   0.576166  -1.378    0.169    
MORT.l1 -3.406546   2.612830  -1.304    0.193    
const    3.647914  40.636318   0.090    0.928    
trend    0.326185   0.271960   1.199    0.231    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 57.19 on 617 degrees of freedom
Multiple R-Squared: 0.9999, Adjusted R-squared: 0.9999 
F-statistic: 7.81e+05 on 6 and 617 DF,  p-value: < 2.2e-16 


Estimation results for equation UMP: 
==================================== 
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1  -4.876e-03  1.470e-02  -0.332   0.7402    
GDP.l1  -1.039e-04  4.488e-05  -2.315   0.0209 *  
UMP.l1   9.392e-01  1.417e-02  66.291   <2e-16 ***
CPI.l1  -2.081e-03  4.689e-03  -0.444   0.6573    
MORT.l1  1.376e-02  2.127e-02   0.647   0.5177    
const    8.324e-01  3.307e-01   2.517   0.0121 *  
trend    3.558e-03  2.213e-03   1.608   0.1085    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4655 on 617 degrees of freedom
Multiple R-Squared: 0.9261, Adjusted R-squared: 0.9254 
F-statistic:  1289 on 6 and 617 DF,  p-value: < 2.2e-16 


Estimation results for equation CPI: 
==================================== 
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   5.849e-02  1.559e-02   3.753 0.000191 ***
GDP.l1   2.482e-04  4.759e-05   5.215 2.51e-07 ***
UMP.l1   3.528e-02  1.502e-02   2.348 0.019168 *  
CPI.l1   1.016e+00  4.972e-03 204.346  < 2e-16 ***
MORT.l1 -2.937e-02  2.255e-02  -1.303 0.193200    
const   -1.834e+00  3.507e-01  -5.229 2.34e-07 ***
trend   -1.194e-02  2.347e-03  -5.085 4.87e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4936 on 617 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared: 0.9999 
F-statistic: 2.073e+06 on 6 and 617 DF,  p-value: < 2.2e-16 


Estimation results for equation MORT: 
===================================== 
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   4.510e-02  8.448e-03   5.339 1.32e-07 ***
GDP.l1   1.213e-05  2.580e-05   0.470   0.6385    
UMP.l1   5.126e-03  8.143e-03   0.629   0.5293    
CPI.l1   5.699e-03  2.695e-03   2.114   0.0349 *  
MORT.l1  9.366e-01  1.222e-02  76.624  < 2e-16 ***
const    2.334e-02  1.901e-01   0.123   0.9023    
trend   -2.683e-03  1.272e-03  -2.109   0.0354 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2675 on 617 degrees of freedom
Multiple R-Squared: 0.9934, Adjusted R-squared: 0.9933 
F-statistic: 1.543e+04 on 6 and 617 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
          FFR       GDP        UMP      CPI      MORT
FFR   0.28161    3.9457  -0.031377  0.02583  0.068411
GDP   3.94574 3270.6260 -14.698226  5.69067  0.482888
UMP  -0.03138  -14.6982   0.216650 -0.05230 -0.007939
CPI   0.02583    5.6907  -0.052298  0.24360  0.022234
MORT  0.06841    0.4829  -0.007939  0.02223  0.071575

Correlation matrix of residuals:
          FFR      GDP      UMP      CPI     MORT
FFR   1.00000  0.13001 -0.12703  0.09863  0.48185
GDP   0.13001  1.00000 -0.55217  0.20161  0.03156
UMP  -0.12703 -0.55217  1.00000 -0.22765 -0.06376
CPI   0.09863  0.20161 -0.22765  1.00000  0.16838
MORT  0.48185  0.03156 -0.06376  0.16838  1.00000


VAR(3)


VAR Estimation Results:
========================= 
Endogenous variables: FFR, GDP, UMP, CPI, MORT 
Deterministic variables: both 
Sample size: 622 
Log Likelihood: -3937.128 
Roots of the characteristic polynomial:
0.9999 0.9999 0.9794 0.9794 0.8771 0.6467 0.6467 0.5721 0.5721 0.4166 0.4166 0.3605 0.3605 0.2805 0.2805
Call:
vars::VAR(y = ts_combined, p = 3, type = "both")


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   1.4252924  0.0447353  31.861  < 2e-16 ***
GDP.l1   0.0007745  0.0005562   1.393  0.16426    
UMP.l1  -0.0143692  0.0517842  -0.277  0.78151    
CPI.l1   0.0255165  0.0454245   0.562  0.57450    
MORT.l1  0.0401763  0.0886834   0.453  0.65069    
FFR.l2  -0.5852555  0.0715553  -8.179 1.69e-15 ***
GDP.l2  -0.0007781  0.0011277  -0.690  0.49044    
UMP.l2  -0.0100167  0.0772807  -0.130  0.89691    
CPI.l2   0.0004464  0.0776948   0.006  0.99542    
MORT.l2 -0.2413403  0.1387852  -1.739  0.08255 .  
FFR.l3   0.1087544  0.0455998   2.385  0.01739 *  
GDP.l3   0.0000650  0.0006748   0.096  0.92329    
UMP.l3  -0.0077867  0.0667742  -0.117  0.90721    
CPI.l3  -0.0256091  0.0468321  -0.547  0.58470    
MORT.l3  0.2412074  0.0855309   2.820  0.00496 ** 
const   -0.0424590  0.3558114  -0.119  0.90505    
trend   -0.0022319  0.0024551  -0.909  0.36368    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4738 on 605 degrees of freedom
Multiple R-Squared: 0.9862, Adjusted R-squared: 0.9858 
F-statistic:  2697 on 16 and 605 DF,  p-value: < 2.2e-16 


Estimation results for equation GDP: 
==================================== 
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend 

         Estimate Std. Error t value Pr(>|t|)    
FFR.l1    6.78267    3.28454   2.065   0.0393 *  
GDP.l1    1.84556    0.04084  45.195  < 2e-16 ***
UMP.l1   79.73754    3.80208  20.972  < 2e-16 ***
CPI.l1   -2.80974    3.33514  -0.842   0.3999    
MORT.l1   5.86310    6.51128   0.900   0.3682    
FFR.l2   -9.18057    5.25371  -1.747   0.0811 .  
GDP.l2   -0.71591    0.08279  -8.647  < 2e-16 ***
UMP.l2  -54.51339    5.67408  -9.607  < 2e-16 ***
CPI.l2    3.82624    5.70448   0.671   0.5026    
MORT.l2  -4.91168   10.18984  -0.482   0.6300    
FFR.l3    1.31237    3.34801   0.392   0.6952    
GDP.l3   -0.13406    0.04954  -2.706   0.0070 ** 
UMP.l3  -23.55930    4.90267  -4.805 1.95e-06 ***
CPI.l3   -1.01128    3.43849  -0.294   0.7688    
MORT.l3  -1.38550    6.27982  -0.221   0.8255    
const    22.81706   26.12426   0.873   0.3828    
trend     0.10449    0.18026   0.580   0.5624    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 34.79 on 605 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 7.861e+05 on 16 and 605 DF,  p-value: < 2.2e-16 


Estimation results for equation UMP: 
==================================== 
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1  -0.0843618  0.0331153  -2.548  0.01110 *  
GDP.l1  -0.0025889  0.0004117  -6.288 6.16e-10 ***
UMP.l1   0.5057201  0.0383332  13.193  < 2e-16 ***
CPI.l1  -0.0438201  0.0336255  -1.303  0.19301    
MORT.l1  0.0312865  0.0656479   0.477  0.63383    
FFR.l2   0.1236302  0.0529688   2.334  0.01992 *  
GDP.l2  -0.0026738  0.0008347  -3.203  0.00143 ** 
UMP.l2   0.0604795  0.0572070   1.057  0.29084    
CPI.l2   0.0202873  0.0575136   0.353  0.72441    
MORT.l2 -0.1050433  0.1027357  -1.022  0.30697    
FFR.l3  -0.0289493  0.0337553  -0.858  0.39144    
GDP.l3   0.0052247  0.0004995  10.460  < 2e-16 ***
UMP.l3   0.4191785  0.0494296   8.480  < 2e-16 ***
CPI.l3   0.0176832  0.0346675   0.510  0.61018    
MORT.l3  0.0738715  0.0633142   1.167  0.24377    
const    0.5413245  0.2633894   2.055  0.04029 *  
trend    0.0036114  0.0018174   1.987  0.04736 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.3507 on 605 degrees of freedom
Multiple R-Squared: 0.9589, Adjusted R-squared: 0.9578 
F-statistic: 881.3 on 16 and 605 DF,  p-value: < 2.2e-16 


Estimation results for equation CPI: 
==================================== 
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   0.0590672  0.0400603   1.474 0.140878    
GDP.l1   0.0016346  0.0004981   3.282 0.001090 ** 
UMP.l1   0.1635305  0.0463725   3.526 0.000453 ***
CPI.l1   1.5014443  0.0406774  36.911  < 2e-16 ***
MORT.l1  0.0891795  0.0794156   1.123 0.261905    
FFR.l2  -0.0502386  0.0640775  -0.784 0.433330    
GDP.l2  -0.0006613  0.0010098  -0.655 0.512788    
UMP.l2  -0.0721825  0.0692046  -1.043 0.297350    
CPI.l2  -0.5992755  0.0695754  -8.613  < 2e-16 ***
MORT.l2 -0.0699906  0.1242816  -0.563 0.573534    
FFR.l3   0.0245324  0.0408344   0.601 0.548213    
GDP.l3  -0.0008465  0.0006043  -1.401 0.161739    
UMP.l3  -0.0773094  0.0597960  -1.293 0.196544    
CPI.l3   0.1054326  0.0419380   2.514 0.012195 *  
MORT.l3 -0.0343605  0.0765926  -0.449 0.653871    
const   -0.9311386  0.3186277  -2.922 0.003604 ** 
trend   -0.0058320  0.0021985  -2.653 0.008196 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4243 on 605 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 1.042e+06 on 16 and 605 DF,  p-value: < 2.2e-16 


Estimation results for equation MORT: 
===================================== 
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   0.1167328  0.0217427   5.369 1.13e-07 ***
GDP.l1  -0.0002459  0.0002703  -0.910 0.363419    
UMP.l1  -0.0026104  0.0251687  -0.104 0.917430    
CPI.l1   0.0860609  0.0220777   3.898 0.000108 ***
MORT.l1  1.3691950  0.0431028  31.766  < 2e-16 ***
FFR.l2  -0.1148019  0.0347780  -3.301 0.001020 ** 
GDP.l2   0.0005009  0.0005481   0.914 0.361074    
UMP.l2   0.0236069  0.0375608   0.628 0.529915    
CPI.l2  -0.1071027  0.0377620  -2.836 0.004717 ** 
MORT.l2 -0.7029532  0.0674538 -10.421  < 2e-16 ***
FFR.l3   0.0287427  0.0221629   1.297 0.195165    
GDP.l3  -0.0002602  0.0003280  -0.793 0.427917    
UMP.l3  -0.0173118  0.0324543  -0.533 0.593939    
CPI.l3   0.0240060  0.0227618   1.055 0.292001    
MORT.l3  0.2872903  0.0415706   6.911 1.22e-11 ***
const    0.1286240  0.1729351   0.744 0.457304    
trend   -0.0011842  0.0011933  -0.992 0.321408    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2303 on 605 degrees of freedom
Multiple R-Squared: 0.9952, Adjusted R-squared: 0.9951 
F-statistic:  7823 on 16 and 605 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
          FFR       GDP       UMP       CPI      MORT
FFR   0.22446    1.2807 -0.011102  0.012258  0.045747
GDP   1.28065 1210.0067 -2.213729  0.598938  0.202303
UMP  -0.01110   -2.2137  0.122998 -0.008206 -0.003838
CPI   0.01226    0.5989 -0.008206  0.179998  0.008554
MORT  0.04575    0.2023 -0.003838  0.008554  0.053023

Correlation matrix of residuals:
          FFR      GDP      UMP      CPI     MORT
FFR   1.00000  0.07771 -0.06682  0.06098  0.41933
GDP   0.07771  1.00000 -0.18146  0.04058  0.02526
UMP  -0.06682 -0.18146  1.00000 -0.05515 -0.04753
CPI   0.06098  0.04058 -0.05515  1.00000  0.08756
MORT  0.41933  0.02526 -0.04753  0.08756  1.00000


VAR(7)


VAR Estimation Results:
========================= 
Endogenous variables: FFR, GDP, UMP, CPI, MORT 
Deterministic variables: both 
Sample size: 618 
Log Likelihood: -3688.661 
Roots of the characteristic polynomial:
0.9951 0.9951 0.9856 0.9446 0.9446 0.8244 0.8244 0.7781 0.7779 0.7779 0.7754 0.7754 0.7484 0.7484 0.7371 0.7371 0.7304 0.7304 0.7228 0.7228 0.6688 0.6688 0.6663 0.6663 0.6441 0.6441  0.64  0.64 0.6361 0.6166 0.6166 0.5937 0.5937 0.3452 0.0158
Call:
vars::VAR(y = ts_combined, p = 7, type = "both")


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   1.403e+00  4.535e-02  30.937  < 2e-16 ***
GDP.l1   8.629e-04  6.405e-04   1.347 0.178400    
UMP.l1  -2.535e-02  6.029e-02  -0.420 0.674308    
CPI.l1   7.933e-03  4.614e-02   0.172 0.863540    
MORT.l1  1.262e-01  9.338e-02   1.352 0.177041    
FFR.l2  -5.487e-01  7.527e-02  -7.290 1.02e-12 ***
GDP.l2  -7.138e-04  1.387e-03  -0.515 0.606971    
UMP.l2  -9.532e-03  8.487e-02  -0.112 0.910620    
CPI.l2  -3.397e-03  8.341e-02  -0.041 0.967528    
MORT.l2 -4.680e-01  1.585e-01  -2.951 0.003290 ** 
FFR.l3   8.124e-02  7.854e-02   1.034 0.301412    
GDP.l3  -1.647e-04  1.489e-03  -0.111 0.911958    
UMP.l3  -7.866e-02  9.223e-02  -0.853 0.394116    
CPI.l3   3.224e-02  8.898e-02   0.362 0.717262    
MORT.l3  6.342e-01  1.710e-01   3.707 0.000229 ***
FFR.l4  -1.004e-01  7.847e-02  -1.279 0.201268    
GDP.l4   1.490e-04  1.443e-03   0.103 0.917793    
UMP.l4  -1.078e-02  9.569e-02  -0.113 0.910357    
CPI.l4  -9.002e-02  8.983e-02  -1.002 0.316741    
MORT.l4 -3.667e-01  1.743e-01  -2.104 0.035779 *  
FFR.l5   1.911e-01  7.844e-02   2.437 0.015124 *  
GDP.l5  -5.456e-04  1.414e-03  -0.386 0.699755    
UMP.l5   3.382e-02  9.109e-02   0.371 0.710572    
CPI.l5   1.196e-01  8.964e-02   1.334 0.182642    
MORT.l5  4.568e-02  1.719e-01   0.266 0.790513    
FFR.l6  -5.761e-02  7.600e-02  -0.758 0.448764    
GDP.l6  -2.237e-05  1.404e-03  -0.016 0.987297    
UMP.l6   4.311e-02  9.141e-02   0.472 0.637371    
CPI.l6  -8.351e-03  8.485e-02  -0.098 0.921630    
MORT.l6 -2.441e-01  1.568e-01  -1.556 0.120146    
FFR.l7  -7.134e-03  4.647e-02  -0.153 0.878061    
GDP.l7   4.989e-04  7.926e-04   0.630 0.529269    
UMP.l7   1.788e-02  7.342e-02   0.243 0.807734    
CPI.l7  -6.016e-02  4.819e-02  -1.248 0.212455    
MORT.l7  3.022e-01  8.997e-02   3.359 0.000834 ***
const    1.501e-02  3.852e-01   0.039 0.968940    
trend   -1.314e-03  2.803e-03  -0.469 0.639477    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.468 on 581 degrees of freedom
Multiple R-Squared: 0.987,  Adjusted R-squared: 0.9862 
F-statistic:  1229 on 36 and 581 DF,  p-value: < 2.2e-16 


Estimation results for equation GDP: 
==================================== 
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend 

         Estimate Std. Error t value Pr(>|t|)    
FFR.l1    5.76923    2.95816   1.950  0.05162 .  
GDP.l1    1.93459    0.04178  46.307  < 2e-16 ***
UMP.l1   75.04375    3.93310  19.080  < 2e-16 ***
CPI.l1   -2.35828    3.00949  -0.784  0.43358    
MORT.l1   4.15780    6.09132   0.683  0.49514    
FFR.l2   -6.94597    4.91022  -1.415  0.15772    
GDP.l2   -0.72973    0.09046  -8.067 4.16e-15 ***
UMP.l2  -56.83715    5.53646 -10.266  < 2e-16 ***
CPI.l2    2.86289    5.44074   0.526  0.59895    
MORT.l2  -5.27329   10.34232  -0.510  0.61033    
FFR.l3   -0.28374    5.12349  -0.055  0.95586    
GDP.l3   -0.58638    0.09712  -6.037 2.79e-09 ***
UMP.l3  -16.86477    6.01635  -2.803  0.00523 ** 
CPI.l3    2.68180    5.80457   0.462  0.64424    
MORT.l3  -5.46856   11.15763  -0.490  0.62424    
FFR.l4    2.02242    5.11884   0.395  0.69292    
GDP.l4    0.63387    0.09413   6.734 3.98e-11 ***
UMP.l4  -11.54743    6.24191  -1.850  0.06482 .  
CPI.l4   -2.20769    5.85982  -0.377  0.70650    
MORT.l4   8.63564   11.36692   0.760  0.44773    
FFR.l5    4.36099    5.11679   0.852  0.39440    
GDP.l5   -0.23234    0.09223  -2.519  0.01204 *  
UMP.l5   14.07096    5.94222   2.368  0.01821 *  
CPI.l5   -0.75688    5.84728  -0.129  0.89705    
MORT.l5 -11.46292   11.21196  -1.022  0.30703    
FFR.l6   -6.59347    4.95742  -1.330  0.18403    
GDP.l6   -0.11159    0.09161  -1.218  0.22366    
UMP.l6   -3.67265    5.96291  -0.616  0.53819    
CPI.l6    1.38198    5.53455   0.250  0.80291    
MORT.l6  14.31850   10.23015   1.400  0.16216    
FFR.l7   -0.24341    3.03158  -0.080  0.93603    
GDP.l7    0.08682    0.05170   1.679  0.09362 .  
UMP.l7    1.36762    4.78947   0.286  0.77533    
CPI.l7   -1.92011    3.14370  -0.611  0.54158    
MORT.l7  -4.16519    5.86874  -0.710  0.47816    
const    29.43663   25.12697   1.172  0.24187    
trend     0.23645    0.18286   1.293  0.19650    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 30.53 on 581 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 4.472e+05 on 36 and 581 DF,  p-value: < 2.2e-16 


Estimation results for equation UMP: 
==================================== 
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1  -0.0852298  0.0297306  -2.867  0.00430 ** 
GDP.l1  -0.0016457  0.0004199  -3.919 9.93e-05 ***
UMP.l1   0.6233340  0.0395291  15.769  < 2e-16 ***
CPI.l1  -0.0803553  0.0302465  -2.657  0.00811 ** 
MORT.l1  0.0280420  0.0612200   0.458  0.64709    
FFR.l2   0.1480852  0.0493496   3.001  0.00281 ** 
GDP.l2  -0.0076606  0.0009092  -8.426 2.82e-16 ***
UMP.l2  -0.0402534  0.0556435  -0.723  0.46972    
CPI.l2   0.1115220  0.0546815   2.039  0.04185 *  
MORT.l2 -0.0397482  0.1039443  -0.382  0.70231    
FFR.l3  -0.0771414  0.0514930  -1.498  0.13465    
GDP.l3   0.0148842  0.0009761  15.248  < 2e-16 ***
UMP.l3   0.7253082  0.0604666  11.995  < 2e-16 ***
CPI.l3  -0.1080739  0.0583381  -1.853  0.06445 .  
MORT.l3  0.0165723  0.1121384   0.148  0.88256    
FFR.l4   0.0077366  0.0514463   0.150  0.88051    
GDP.l4  -0.0051300  0.0009460  -5.423 8.63e-08 ***
UMP.l4  -0.3493726  0.0627335  -5.569 3.92e-08 ***
CPI.l4   0.1228308  0.0588934   2.086  0.03745 *  
MORT.l4  0.0592520  0.1142418   0.519  0.60420    
FFR.l5  -0.0298678  0.0514257  -0.581  0.56160    
GDP.l5  -0.0053343  0.0009270  -5.754 1.41e-08 ***
UMP.l5  -0.0161097  0.0597216  -0.270  0.78745    
CPI.l5  -0.0740392  0.0587674  -1.260  0.20822    
MORT.l5 -0.1108013  0.1126845  -0.983  0.32587    
FFR.l6   0.0398457  0.0498240   0.800  0.42419    
GDP.l6   0.0077572  0.0009207   8.425 2.83e-16 ***
UMP.l6   0.0355911  0.0599295   0.594  0.55282    
CPI.l6   0.0177462  0.0556243   0.319  0.74981    
MORT.l6  0.0585746  0.1028169   0.570  0.56910    
FFR.l7   0.0110064  0.0304685   0.361  0.71805    
GDP.l7  -0.0028968  0.0005196  -5.575 3.79e-08 ***
UMP.l7   0.0050475  0.0481359   0.105  0.91652    
CPI.l7   0.0073505  0.0315953   0.233  0.81612    
MORT.l7 -0.0190374  0.0589831  -0.323  0.74699    
const    0.4379432  0.2525355   1.734  0.08342 .  
trend    0.0021652  0.0018378   1.178  0.23923    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.3068 on 581 degrees of freedom
Multiple R-Squared: 0.9698, Adjusted R-squared: 0.9679 
F-statistic: 517.5 on 36 and 581 DF,  p-value: < 2.2e-16 


Estimation results for equation CPI: 
==================================== 
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   0.0361597  0.0407843   0.887  0.37566    
GDP.l1   0.0013010  0.0005760   2.259  0.02427 *  
UMP.l1   0.1713298  0.0542257   3.160  0.00166 ** 
CPI.l1   1.4993723  0.0414919  36.136  < 2e-16 ***
MORT.l1  0.1052732  0.0839812   1.254  0.21052    
FFR.l2  -0.0094818  0.0676974  -0.140  0.88866    
GDP.l2   0.0008479  0.0012472   0.680  0.49688    
UMP.l2  -0.0696796  0.0763313  -0.913  0.36170    
CPI.l2  -0.6585270  0.0750117  -8.779  < 2e-16 ***
MORT.l2 -0.0903743  0.1425900  -0.634  0.52646    
FFR.l3  -0.0218046  0.0706377  -0.309  0.75767    
GDP.l3  -0.0021381  0.0013391  -1.597  0.11087    
UMP.l3  -0.1751041  0.0829477  -2.111  0.03520 *  
CPI.l3   0.2613886  0.0800278   3.266  0.00115 ** 
MORT.l3 -0.0959855  0.1538307  -0.624  0.53289    
FFR.l4   0.0025181  0.0705737   0.036  0.97155    
GDP.l4  -0.0009838  0.0012978  -0.758  0.44872    
UMP.l4  -0.0582700  0.0860574  -0.677  0.49861    
CPI.l4  -0.0115904  0.0807896  -0.143  0.88597    
MORT.l4  0.1565024  0.1567161   0.999  0.31839    
FFR.l5   0.0343926  0.0705455   0.488  0.62607    
GDP.l5   0.0017279  0.0012716   1.359  0.17472    
UMP.l5   0.0461154  0.0819257   0.563  0.57372    
CPI.l5  -0.1216232  0.0806167  -1.509  0.13193    
MORT.l5 -0.2029994  0.1545798  -1.313  0.18962    
FFR.l6   0.0028868  0.0683482   0.042  0.96632    
GDP.l6  -0.0019113  0.0012630  -1.513  0.13074    
UMP.l6  -0.0112452  0.0822109  -0.137  0.89125    
CPI.l6   0.0755170  0.0763051   0.990  0.32275    
MORT.l6  0.0520858  0.1410435   0.369  0.71205    
FFR.l7  -0.0046953  0.0417965  -0.112  0.91059    
GDP.l7   0.0012596  0.0007128   1.767  0.07772 .  
UMP.l7   0.1198452  0.0660326   1.815  0.07005 .  
CPI.l7  -0.0426923  0.0433423  -0.985  0.32503    
MORT.l7  0.0524162  0.0809126   0.648  0.51736    
const   -0.6699990  0.3464265  -1.934  0.05359 .  
trend   -0.0030076  0.0025211  -1.193  0.23337    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4209 on 581 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 4.618e+05 on 36 and 581 DF,  p-value: < 2.2e-16 


Estimation results for equation MORT: 
===================================== 
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1   1.170e-01  2.220e-02   5.270 1.93e-07 ***
GDP.l1  -3.297e-04  3.136e-04  -1.051 0.293534    
UMP.l1   1.576e-02  2.952e-02   0.534 0.593680    
CPI.l1   7.578e-02  2.259e-02   3.355 0.000846 ***
MORT.l1  1.395e+00  4.572e-02  30.508  < 2e-16 ***
FFR.l2  -1.203e-01  3.686e-02  -3.264 0.001163 ** 
GDP.l2   5.030e-04  6.790e-04   0.741 0.459089    
UMP.l2   4.608e-02  4.156e-02   1.109 0.267905    
CPI.l2  -1.152e-01  4.084e-02  -2.821 0.004951 ** 
MORT.l2 -7.614e-01  7.763e-02  -9.808  < 2e-16 ***
FFR.l3   6.682e-02  3.846e-02   1.738 0.082818 .  
GDP.l3   5.567e-04  7.290e-04   0.764 0.445411    
UMP.l3  -3.250e-02  4.516e-02  -0.720 0.471965    
CPI.l3   6.360e-02  4.357e-02   1.460 0.144904    
MORT.l3  3.656e-01  8.375e-02   4.366 1.50e-05 ***
FFR.l4  -8.747e-02  3.842e-02  -2.277 0.023169 *  
GDP.l4  -5.624e-04  7.065e-04  -0.796 0.426325    
UMP.l4  -5.438e-02  4.685e-02  -1.161 0.246269    
CPI.l4   1.744e-02  4.398e-02   0.396 0.691923    
MORT.l4 -1.167e-02  8.532e-02  -0.137 0.891287    
FFR.l5   1.047e-01  3.841e-02   2.726 0.006608 ** 
GDP.l5  -7.861e-04  6.923e-04  -1.135 0.256669    
UMP.l5  -4.963e-02  4.460e-02  -1.113 0.266314    
CPI.l5  -3.875e-02  4.389e-02  -0.883 0.377673    
MORT.l5 -7.099e-02  8.416e-02  -0.844 0.399248    
FFR.l6  -5.433e-02  3.721e-02  -1.460 0.144835    
GDP.l6   4.214e-04  6.876e-04   0.613 0.540226    
UMP.l6   3.897e-02  4.476e-02   0.871 0.384305    
CPI.l6   1.259e-02  4.154e-02   0.303 0.762026    
MORT.l6 -8.260e-04  7.679e-02  -0.011 0.991421    
FFR.l7  -9.717e-05  2.275e-02  -0.004 0.996594    
GDP.l7   1.814e-04  3.881e-04   0.467 0.640338    
UMP.l7   4.285e-02  3.595e-02   1.192 0.233739    
CPI.l7  -1.500e-02  2.360e-02  -0.636 0.525271    
MORT.l7  4.212e-02  4.405e-02   0.956 0.339370    
const    2.204e-01  1.886e-01   1.169 0.243072    
trend    4.001e-05  1.373e-03   0.029 0.976756    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.2291 on 581 degrees of freedom
Multiple R-Squared: 0.9954, Adjusted R-squared: 0.9951 
F-statistic:  3512 on 36 and 581 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
           FFR      GDP       UMP       CPI      MORT
FFR   0.219033   1.3061 -0.010856  0.008961  0.044674
GDP   1.306052 932.0092 -0.995366  0.353180  0.194479
UMP  -0.010856  -0.9954  0.094142 -0.002369 -0.004714
CPI   0.008961   0.3532 -0.002369  0.177159  0.006045
MORT  0.044674   0.1945 -0.004714  0.006045  0.052507

Correlation matrix of residuals:
          FFR      GDP      UMP      CPI     MORT
FFR   1.00000  0.09141 -0.07560  0.04549  0.41657
GDP   0.09141  1.00000 -0.10626  0.02749  0.02780
UMP  -0.07560 -0.10626  1.00000 -0.01834 -0.06705
CPI   0.04549  0.02749 -0.01834  1.00000  0.06267
MORT  0.41657  0.02780 -0.06705  0.06267  1.00000


VAR(10)


VAR Estimation Results:
========================= 
Endogenous variables: FFR, GDP, UMP, CPI, MORT 
Deterministic variables: both 
Sample size: 615 
Log Likelihood: -3563.255 
Roots of the characteristic polynomial:
0.9918 0.9918 0.9786 0.9494 0.9494 0.8982 0.8705 0.8705 0.8669 0.8669 0.8619 0.8619 0.8344 0.8344 0.8339 0.8339 0.8286 0.8286 0.8263 0.8263 0.8261 0.8138 0.8138 0.8059 0.8059 0.8032 0.8032 0.7879 0.7879 0.787 0.787 0.781 0.781 0.7606 0.7606 0.7562 0.7562 0.7546 0.7546 0.7481 0.7481 0.7469 0.7469 0.7386 0.7118 0.7118 0.6778 0.6778 0.6435 0.6435
Call:
vars::VAR(y = ts_combined, p = 10, type = "both")


Estimation results for equation FFR: 
==================================== 
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    1.3925124  0.0456139  30.528  < 2e-16 ***
GDP.l1    0.0006535  0.0006170   1.059 0.290016    
UMP.l1   -0.0249535  0.0631346  -0.395 0.692814    
CPI.l1    0.0146835  0.0443089   0.331 0.740474    
MORT.l1   0.1577118  0.0903379   1.746 0.081391 .  
FFR.l2   -0.5144433  0.0756368  -6.801 2.65e-11 ***
GDP.l2   -0.0002373  0.0013374  -0.177 0.859261    
UMP.l2    0.0508775  0.0889707   0.572 0.567655    
CPI.l2   -0.0097948  0.0801393  -0.122 0.902767    
MORT.l2  -0.4677727  0.1539358  -3.039 0.002486 ** 
FFR.l3    0.0750617  0.0783067   0.959 0.338192    
GDP.l3   -0.0006841  0.0015109  -0.453 0.650883    
UMP.l3   -0.1413837  0.0945131  -1.496 0.135236    
CPI.l3    0.0260103  0.0860013   0.302 0.762427    
MORT.l3   0.4901616  0.1659450   2.954 0.003270 ** 
FFR.l4   -0.1161904  0.0765430  -1.518 0.129582    
GDP.l4    0.0009027  0.0018361   0.492 0.623173    
UMP.l4    0.0293343  0.1021184   0.287 0.774021    
CPI.l4   -0.0865638  0.0868835  -0.996 0.319523    
MORT.l4  -0.2423112  0.1678998  -1.443 0.149523    
FFR.l5    0.2098276  0.0760056   2.761 0.005956 ** 
GDP.l5   -0.0016801  0.0019176  -0.876 0.381324    
UMP.l5    0.0425428  0.1039950   0.409 0.682633    
CPI.l5    0.1195584  0.0873215   1.369 0.171490    
MORT.l5   0.0877093  0.1685299   0.520 0.602963    
FFR.l6    0.0125900  0.0763905   0.165 0.869152    
GDP.l6    0.0009697  0.0018917   0.513 0.608412    
UMP.l6    0.0681907  0.1031392   0.661 0.508785    
CPI.l6   -0.0549973  0.0874025  -0.629 0.529446    
MORT.l6  -0.4285943  0.1685841  -2.542 0.011279 *  
FFR.l7   -0.2918106  0.0763471  -3.822 0.000147 ***
GDP.l7    0.0015311  0.0016305   0.939 0.348116    
UMP.l7   -0.0058253  0.0952556  -0.061 0.951258    
CPI.l7    0.0544967  0.0871326   0.625 0.531932    
MORT.l7   0.8164602  0.1696519   4.813 1.92e-06 ***
FFR.l8    0.3174611  0.0772681   4.109 4.57e-05 ***
GDP.l8   -0.0023407  0.0015428  -1.517 0.129791    
UMP.l8   -0.0980588  0.0880199  -1.114 0.265732    
CPI.l8   -0.1037529  0.0863820  -1.201 0.230220    
MORT.l8  -0.3277054  0.1704243  -1.923 0.054999 .  
FFR.l9   -0.0539941  0.0765108  -0.706 0.480663    
GDP.l9    0.0005875  0.0014808   0.397 0.691708    
UMP.l9    0.0037559  0.0880419   0.043 0.965987    
CPI.l9    0.0725219  0.0813894   0.891 0.373284    
MORT.l9   0.0281445  0.1594137   0.177 0.859926    
FFR.l10  -0.0880616  0.0462636  -1.903 0.057489 .  
GDP.l10   0.0003370  0.0007986   0.422 0.673197    
UMP.l10   0.0550137  0.0708891   0.776 0.438043    
CPI.l10  -0.0341610  0.0465982  -0.733 0.463805    
MORT.l10 -0.0768781  0.0912497  -0.843 0.399865    
const     0.1470506  0.3953724   0.372 0.710085    
trend    -0.0008793  0.0030032  -0.293 0.769798    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.4456 on 563 degrees of freedom
Multiple R-Squared: 0.9886, Adjusted R-squared: 0.9876 
F-statistic: 958.8 on 51 and 563 DF,  p-value: < 2.2e-16 


Estimation results for equation GDP: 
==================================== 
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend 

          Estimate Std. Error t value Pr(>|t|)    
FFR.l1     4.98895    3.11725   1.600  0.11006    
GDP.l1     1.93129    0.04217  45.799  < 2e-16 ***
UMP.l1    73.04305    4.31460  16.929  < 2e-16 ***
CPI.l1    -2.46602    3.02806  -0.814  0.41577    
MORT.l1    4.06364    6.17367   0.658  0.51067    
FFR.l2    -4.81802    5.16901  -0.932  0.35169    
GDP.l2    -0.71996    0.09140  -7.877 1.74e-14 ***
UMP.l2   -52.37044    6.08024  -8.613  < 2e-16 ***
CPI.l2     3.24432    5.47671   0.592  0.55383    
MORT.l2   -3.58085   10.51995  -0.340  0.73369    
FFR.l3    -0.99520    5.35147  -0.186  0.85254    
GDP.l3    -0.64714    0.10325  -6.268 7.30e-10 ***
UMP.l3   -17.53285    6.45901  -2.714  0.00684 ** 
CPI.l3     2.60111    5.87731   0.443  0.65825    
MORT.l3   -9.74289   11.34065  -0.859  0.39064    
FFR.l4     2.09603    5.23094   0.401  0.68879    
GDP.l4     0.76221    0.12548   6.075 2.29e-09 ***
UMP.l4    -9.31023    6.97875  -1.334  0.18272    
CPI.l4    -2.71479    5.93760  -0.457  0.64769    
MORT.l4    9.10700   11.47424   0.794  0.42771    
FFR.l5     4.23222    5.19421   0.815  0.41553    
GDP.l5    -0.29789    0.13105  -2.273  0.02339 *  
UMP.l5    12.54793    7.10700   1.766  0.07801 .  
CPI.l5     0.17272    5.96753   0.029  0.97692    
MORT.l5   -6.16596   11.51730  -0.535  0.59261    
FFR.l6    -6.44129    5.22051  -1.234  0.21778    
GDP.l6    -0.19456    0.12928  -1.505  0.13290    
UMP.l6    -4.13621    7.04851  -0.587  0.55756    
CPI.l6    -0.75880    5.97307  -0.127  0.89896    
MORT.l6    7.27257   11.52100   0.631  0.52814    
FFR.l7    -0.47794    5.21755  -0.092  0.92705    
GDP.l7     0.27372    0.11143   2.457  0.01433 *  
UMP.l7     2.63440    6.50975   0.405  0.68586    
CPI.l7     0.69867    5.95462   0.117  0.90664    
MORT.l7    3.90808   11.59398   0.337  0.73618    
FFR.l8    -0.13376    5.28049  -0.025  0.97980    
GDP.l8    -0.11138    0.10544  -1.056  0.29124    
UMP.l8    -3.52894    6.01526  -0.587  0.55766    
CPI.l8     4.16021    5.90333   0.705  0.48127    
MORT.l8   -0.25103   11.64676  -0.022  0.98281    
FFR.l9     4.82701    5.22873   0.923  0.35631    
GDP.l9    -0.05080    0.10120  -0.502  0.61587    
UMP.l9    -2.09553    6.01676  -0.348  0.72776    
CPI.l9    -9.39502    5.56214  -1.689  0.09175 .  
MORT.l9   -6.02749   10.89430  -0.553  0.58030    
FFR.l10   -6.09321    3.16165  -1.927  0.05445 .  
GDP.l10    0.04899    0.05457   0.898  0.36969    
UMP.l10    2.40878    4.84455   0.497  0.61923    
CPI.l10    4.06839    3.18451   1.278  0.20193    
MORT.l10   3.11147    6.23598   0.499  0.61801    
const     30.62362   27.01967   1.133  0.25754    
trend      0.28405    0.20524   1.384  0.16690    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 30.45 on 563 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 3.139e+05 on 51 and 563 DF,  p-value: < 2.2e-16 


Estimation results for equation UMP: 
==================================== 
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1   -0.0962662  0.0295834  -3.254 0.001206 ** 
GDP.l1   -0.0015077  0.0004002  -3.768 0.000182 ***
UMP.l1    0.6983075  0.0409466  17.054  < 2e-16 ***
CPI.l1   -0.0807023  0.0287370  -2.808 0.005153 ** 
MORT.l1   0.0190020  0.0585896   0.324 0.745813    
FFR.l2    0.1580754  0.0490551   3.222 0.001345 ** 
GDP.l2   -0.0082853  0.0008674  -9.552  < 2e-16 ***
UMP.l2   -0.0554728  0.0577029  -0.961 0.336788    
CPI.l2    0.1157673  0.0519752   2.227 0.026318 *  
MORT.l2  -0.0112406  0.0998367  -0.113 0.910396    
FFR.l3   -0.0778622  0.0507867  -1.533 0.125808    
GDP.l3    0.0172790  0.0009799  17.634  < 2e-16 ***
UMP.l3    0.6025063  0.0612975   9.829  < 2e-16 ***
CPI.l3   -0.0980851  0.0557770  -1.759 0.079202 .  
MORT.l3  -0.0077874  0.1076254  -0.072 0.942344    
FFR.l4   -0.0072534  0.0496428  -0.146 0.883886    
GDP.l4   -0.0072583  0.0011908  -6.095 2.03e-09 ***
UMP.l4   -0.3539175  0.0662300  -5.344 1.32e-07 ***
CPI.l4    0.0905868  0.0563493   1.608 0.108484    
MORT.l4   0.0794237  0.1088932   0.729 0.466077    
FFR.l5   -0.0048509  0.0492942  -0.098 0.921644    
GDP.l5   -0.0087046  0.0012437  -6.999 7.36e-12 ***
UMP.l5   -0.0701354  0.0674471  -1.040 0.298852    
CPI.l5   -0.0427311  0.0566333  -0.755 0.450851    
MORT.l5  -0.1564678  0.1093019  -1.432 0.152836    
FFR.l6    0.0176201  0.0495439   0.356 0.722239    
GDP.l6    0.0147355  0.0012269  12.010  < 2e-16 ***
UMP.l6    0.2278423  0.0668921   3.406 0.000706 ***
CPI.l6    0.0137721  0.0566858   0.243 0.808128    
MORT.l6   0.1375262  0.1093370   1.258 0.208979    
FFR.l7    0.0779449  0.0495158   1.574 0.116016    
GDP.l7   -0.0066642  0.0010575  -6.302 5.93e-10 ***
UMP.l7   -0.0675298  0.0617791  -1.093 0.274824    
CPI.l7   -0.0141871  0.0565108  -0.251 0.801866    
MORT.l7  -0.1480836  0.1100296  -1.346 0.178892    
FFR.l8   -0.1522983  0.0501130  -3.039 0.002483 ** 
GDP.l8   -0.0030433  0.0010006  -3.041 0.002465 ** 
UMP.l8    0.0157603  0.0570862   0.276 0.782589    
CPI.l8    0.0040020  0.0560240   0.071 0.943079    
MORT.l8   0.1079815  0.1105305   0.977 0.329019    
FFR.l9    0.1189025  0.0496219   2.396 0.016893 *  
GDP.l9    0.0061626  0.0009604   6.417 2.96e-10 ***
UMP.l9    0.0614381  0.0571005   1.076 0.282404    
CPI.l9    0.0246464  0.0527860   0.467 0.640744    
MORT.l9   0.0333963  0.1033895   0.323 0.746804    
FFR.l10  -0.0188796  0.0300048  -0.629 0.529460    
GDP.l10  -0.0027436  0.0005179  -5.298 1.69e-07 ***
UMP.l10  -0.0771065  0.0459759  -1.677 0.094076 .  
CPI.l10  -0.0166136  0.0302217  -0.550 0.582727    
MORT.l10 -0.0627474  0.0591810  -1.060 0.289479    
const     0.4920408  0.2564230   1.919 0.055507 .  
trend     0.0024495  0.0019477   1.258 0.209052    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.289 on 563 degrees of freedom
Multiple R-Squared: 0.974,  Adjusted R-squared: 0.9716 
F-statistic: 413.6 on 51 and 563 DF,  p-value: < 2.2e-16 


Estimation results for equation CPI: 
==================================== 
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    3.907e-02  4.331e-02   0.902  0.36735    
GDP.l1    1.294e-03  5.859e-04   2.208  0.02763 *  
UMP.l1    1.600e-01  5.994e-02   2.669  0.00782 ** 
CPI.l1    1.502e+00  4.207e-02  35.703  < 2e-16 ***
MORT.l1   1.294e-01  8.577e-02   1.508  0.13203    
FFR.l2   -3.812e-02  7.181e-02  -0.531  0.59574    
GDP.l2    8.255e-04  1.270e-03   0.650  0.51589    
UMP.l2   -2.796e-02  8.447e-02  -0.331  0.74074    
CPI.l2   -6.645e-01  7.609e-02  -8.734  < 2e-16 ***
MORT.l2  -1.235e-01  1.462e-01  -0.845  0.39847    
FFR.l3    2.957e-02  7.435e-02   0.398  0.69098    
GDP.l3   -2.246e-03  1.435e-03  -1.566  0.11797    
UMP.l3   -2.336e-01  8.974e-02  -2.603  0.00948 ** 
CPI.l3    2.646e-01  8.166e-02   3.240  0.00127 ** 
MORT.l3  -1.097e-01  1.576e-01  -0.696  0.48651    
FFR.l4   -2.956e-02  7.268e-02  -0.407  0.68435    
GDP.l4   -2.533e-04  1.743e-03  -0.145  0.88451    
UMP.l4   -4.678e-03  9.696e-02  -0.048  0.96154    
CPI.l4   -2.182e-02  8.249e-02  -0.264  0.79149    
MORT.l4   1.778e-01  1.594e-01   1.115  0.26524    
FFR.l5    4.774e-02  7.216e-02   0.662  0.50849    
GDP.l5    9.014e-05  1.821e-03   0.050  0.96053    
UMP.l5    9.037e-03  9.874e-02   0.092  0.92711    
CPI.l5   -1.034e-01  8.291e-02  -1.247  0.21287    
MORT.l5  -1.870e-01  1.600e-01  -1.168  0.24313    
FFR.l6   -5.537e-04  7.253e-02  -0.008  0.99391    
GDP.l6   -2.867e-04  1.796e-03  -0.160  0.87326    
UMP.l6    6.769e-02  9.793e-02   0.691  0.48974    
CPI.l6    2.153e-02  8.299e-02   0.259  0.79536    
MORT.l6  -1.278e-02  1.601e-01  -0.080  0.93637    
FFR.l7   -3.751e-02  7.249e-02  -0.517  0.60503    
GDP.l7    1.515e-03  1.548e-03   0.979  0.32807    
UMP.l7    1.047e-01  9.044e-02   1.158  0.24737    
CPI.l7    8.547e-02  8.273e-02   1.033  0.30200    
MORT.l7   2.285e-01  1.611e-01   1.418  0.15664    
FFR.l8    7.617e-02  7.336e-02   1.038  0.29962    
GDP.l8   -1.447e-03  1.465e-03  -0.988  0.32369    
UMP.l8   -5.184e-02  8.357e-02  -0.620  0.53529    
CPI.l8   -9.164e-02  8.202e-02  -1.117  0.26435    
MORT.l8  -2.547e-01  1.618e-01  -1.574  0.11607    
FFR.l9   -1.029e-01  7.264e-02  -1.416  0.15720    
GDP.l9    7.744e-04  1.406e-03   0.551  0.58201    
UMP.l9   -5.111e-02  8.359e-02  -0.611  0.54117    
CPI.l9    2.656e-02  7.728e-02   0.344  0.73115    
MORT.l9   2.816e-01  1.514e-01   1.860  0.06334 .  
FFR.l10   5.607e-02  4.393e-02   1.277  0.20228    
GDP.l10  -1.880e-04  7.582e-04  -0.248  0.80422    
UMP.l10   5.368e-02  6.731e-02   0.798  0.42545    
CPI.l10  -1.938e-02  4.424e-02  -0.438  0.66150    
MORT.l10 -1.574e-01  8.664e-02  -1.817  0.06978 .  
const    -4.635e-01  3.754e-01  -1.235  0.21746    
trend    -1.469e-03  2.851e-03  -0.515  0.60670    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.423 on 563 degrees of freedom
Multiple R-Squared:     1,  Adjusted R-squared:     1 
F-statistic: 3.18e+05 on 51 and 563 DF,  p-value: < 2.2e-16 


Estimation results for equation MORT: 
===================================== 
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend 

           Estimate Std. Error t value Pr(>|t|)    
FFR.l1    0.1328152  0.0231328   5.741 1.54e-08 ***
GDP.l1   -0.0003791  0.0003129  -1.211 0.226282    
UMP.l1    0.0048796  0.0320183   0.152 0.878926    
CPI.l1    0.0766326  0.0224710   3.410 0.000695 ***
MORT.l1   1.4076295  0.0458143  30.725  < 2e-16 ***
FFR.l2   -0.1419178  0.0383587  -3.700 0.000237 ***
GDP.l2    0.0006852  0.0006783   1.010 0.312795    
UMP.l2    0.0590239  0.0451209   1.308 0.191364    
CPI.l2   -0.1182545  0.0406421  -2.910 0.003761 ** 
MORT.l2  -0.7692118  0.0780676  -9.853  < 2e-16 ***
FFR.l3    0.0646591  0.0397128   1.628 0.104048    
GDP.l3    0.0001740  0.0007662   0.227 0.820449    
UMP.l3   -0.0530320  0.0479317  -1.106 0.269023    
CPI.l3    0.0596652  0.0436150   1.368 0.171859    
MORT.l3   0.3466754  0.0841579   4.119 4.37e-05 ***
FFR.l4   -0.0839070  0.0388183  -2.162 0.031075 *  
GDP.l4   -0.0001326  0.0009311  -0.142 0.886812    
UMP.l4   -0.0368019  0.0517887  -0.711 0.477616    
CPI.l4    0.0238755  0.0440624   0.542 0.588132    
MORT.l4   0.0234867  0.0851493   0.276 0.782780    
FFR.l5    0.1093545  0.0385457   2.837 0.004718 ** 
GDP.l5   -0.0011464  0.0009725  -1.179 0.238969    
UMP.l5   -0.0633593  0.0527404  -1.201 0.230123    
CPI.l5   -0.0450105  0.0442845  -1.016 0.309879    
MORT.l5  -0.0921020  0.0854689  -1.078 0.281670    
FFR.l6   -0.0200262  0.0387409  -0.517 0.605410    
GDP.l6    0.0005742  0.0009594   0.598 0.549767    
UMP.l6    0.0645909  0.0523064   1.235 0.217399    
CPI.l6    0.0091826  0.0443256   0.207 0.835958    
MORT.l6  -0.0283568  0.0854963  -0.332 0.740260    
FFR.l7   -0.1422721  0.0387190  -3.674 0.000261 ***
GDP.l7    0.0002795  0.0008269   0.338 0.735510    
UMP.l7    0.0160036  0.0483083   0.331 0.740556    
CPI.l7    0.0027166  0.0441887   0.061 0.951001    
MORT.l7   0.2056750  0.0860379   2.391 0.017152 *  
FFR.l8    0.1732977  0.0391860   4.422 1.17e-05 ***
GDP.l8    0.0002608  0.0007824   0.333 0.739011    
UMP.l8    0.0304853  0.0446387   0.683 0.494930    
CPI.l8   -0.0318458  0.0438081  -0.727 0.467566    
MORT.l8  -0.1808582  0.0864296  -2.093 0.036836 *  
FFR.l9   -0.0849790  0.0388020  -2.190 0.028929 *  
GDP.l9   -0.0008728  0.0007510  -1.162 0.245666    
UMP.l9   -0.0730508  0.0446499  -1.636 0.102381    
CPI.l9    0.0678871  0.0412761   1.645 0.100589    
MORT.l9   0.0095171  0.0808456   0.118 0.906332    
FFR.l10   0.0232036  0.0234623   0.989 0.323100    
GDP.l10   0.0005296  0.0004050   1.308 0.191500    
UMP.l10   0.0599008  0.0359509   1.666 0.096234 .  
CPI.l10  -0.0459623  0.0236320  -1.945 0.052282 .  
MORT.l10  0.0302584  0.0462767   0.654 0.513470    
const     0.3377335  0.2005106   1.684 0.092665 .  
trend     0.0009649  0.0015230   0.634 0.526659    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 0.226 on 563 degrees of freedom
Multiple R-Squared: 0.9957, Adjusted R-squared: 0.9953 
F-statistic:  2550 on 51 and 563 DF,  p-value: < 2.2e-16 



Covariance matrix of residuals:
           FFR      GDP       UMP       CPI      MORT
FFR   0.198526   0.8625 -0.007206  0.004755  0.040735
GDP   0.862483 927.1797 -0.745157  0.322888  0.201382
UMP  -0.007206  -0.7452  0.083506 -0.003017 -0.003125
CPI   0.004755   0.3229 -0.003017  0.178968  0.004934
MORT  0.040735   0.2014 -0.003125  0.004934  0.051060

Correlation matrix of residuals:
          FFR      GDP      UMP      CPI     MORT
FFR   1.00000  0.06357 -0.05596  0.02523  0.40460
GDP   0.06357  1.00000 -0.08469  0.02507  0.02927
UMP  -0.05596 -0.08469  1.00000 -0.02468 -0.04785
CPI   0.02523  0.02507 -0.02468  1.00000  0.05161
MORT  0.40460  0.02927 -0.04785  0.05161  1.00000


Hyperparameter Optimization

Model fitting with minimum AIC:
 10, 7646.50982266218, 8796.13161229187

Model fitting with minimum BIC:
 3, 8044.25621302432, 8421.05612090715


Cross Validation

The plots compare the cross-validation RMSE for VAR(3) and VAR(10) models across different financial variables over time. The consistently lower RMSE values for VAR(1) indicate that it provides a better fit or more accurate forecasts for these series.


Forecasting

The VAR(3) model forecasts for key economic indicators suggest a tightening monetary environment, with rising Federal Funds Rates potentially in response to upward inflationary trends indicated by the Consumer Price Index. Consistent economic growth is anticipated, as evidenced by the steady increase in GDP, though the projection comes with increasing uncertainty. The sharp uptick in forecasted 30-Year Mortgage Rates could dampen housing market activity by raising borrowing costs. Meanwhile, the expected rise in the unemployment rate signals possible headwinds for the labor market, which may require targeted economic policies to mitigate the risk of a slowdown and support job growth.